Curve Sketching

The graph of a given function gives a visual presentation of the behaviour of the function involving a study of symmetry, rise and fall, region of existence, passage through specific points etc.

The following points are helpful to trace the curve.

Symmetry

a) Symmetry about x - axis

If the equation of the curve remains unaltered when y is replaced by -y, then the curve is symmetrical about x-axis.

b) Symmetry about y-axis

If the equation of the curve remains unaltered when x is replaced by -x then the curve is symmetrical about y-axis.

c) Symmetry about y = x

If the equation of the curve remains unaltered if x and y are interchanged, then the curve is symmetrical about y = x.

d) Symmetry about y = -x

If x and y are replaced by -y and -x and the equation of the curve is unaltered, then the curve is symmetrical about the line y = -x.

e) Symmetry in opposite quadrants

If the equation of the curve is unaltered, when x and y are replaced by -x and -y, then it is symmetrical in opposite quadrants.

Example:

Sketch the curve

y = - sin 2x ….(1)

Solution:

Symmetry

(a) By replacing y by -y, the equation (1) is altered, therefore the curve is not symmetrical about x-axis.

(b) By replacing x by -x, the equation of the curve is altered, therefore the curve is not symmetrical about y-axis.

(c) Replace x and y by -x and -y respectively in the equation y = - sin 2x, the equation of the curve remain unaltered. Therefore the curve is symmetrical in opposite quadrants.

Passage through origin

Put x = 0, y = -sin2x = 0. This implies (0, 0) is a point on the curve or the curve passes through the origin.

Points of intersection

The points of intersection with x-axis is determined by letting y = 0.

Putting y = 0, - sin 2x = 0

This implies the curve intersects the x-axis at the points where

The points at which the tangent is parallel to x-axes are determined by solving.

y = - sin 2x

The tangent is parallel to x- axis at

We know that sin x is a periodic function with 2p, that is

sin(2p + x) = sinx

The pattern of the curves repeats at an interval p.

Therefore it is sufficient to sketch the curve form 0 to p.

Determination of Few Point

with this information, we sketch the graph as show .